在双环面上嵌入K3、3和K5
Embedding K3,3 and K5 on the Double Torus
摘要：
Kuratowski图$K_{3,3}$和$K_5$表征平面性。计算这两个图在可定向曲面上的不同2-胞嵌入以前是通过使用Burnside引理和它们的自同构群来完成的，而没有实际构造嵌入。我们得到所有的2-细胞嵌入这些图的双环面，使用建设性的方法。这表明在双环面上存在唯一的$K_{3,3}$的不可定向2-胞腔嵌入、$K_5$的14个可定向2-胞腔嵌入和17个不可定向2-胞腔嵌入，这明确地证实了计数结果。因此，提出了双环面的几种新的多边形表示。
英文摘要：
The Kuratowski graphs $K_{3,3}$ and $K_5$ characterize planarity. Counting distinct 2-cell embeddings of these two graphs on orientable surfaces was previously done by using Burnside's Lemma and their automorphism groups, without actually constructing the embeddings. We obtain all 2-cell embeddings of these graphs on the double torus, using a constructive approach. This shows that there is a unique non-orientable 2-cell embedding of $K_{3,3}$, 14 orientable and 17 non-orientable 2-cell embeddings of $K_5$ on the double torus, which explicitly confirms the enumerative results. As a consequence, several new polygonal representations of the double torus are presented.
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